The Numbers Game

By

Keith Devlin

January 28, 2012

Though Sudoku appeared on the scene very recently (first popularized in Japan in the 1980s, it only took off in the West in 2004, when the Times of London began featuring it), it has rapidly become one of the most successful paper-and-pencil puzzles of all time. The 9-by-9 Sudoku grid is now a daily staple of many newspapers, and naturally, in a world where there is an app for almost everything, there are digital versions as well.

The name Sudoku comes from the Japanese Sūji wa dokushin ni kagiru ("the digits are limited to one occurrence," reflecting the rule that each digit must appear once in each row, column and sub-square). The passion and commitment engendered in millions of ordinary people by the challenge of filling in the digits 1 through 9 is so different from the antipathy that many profess toward mathematics that it must surely be the case that, those nine numbers notwithstanding, the puzzle is not a mathematical one. As one commentator, Jean-Paul Delahaye, observed in Scientific American: "Despite being a game of numbers, Sudoku demands not an iota of mathematics of its solvers. In fact, no operation—including addition or multiplication—helps in completing a grid, which in theory could be filled with any set of nine different symbols (letters, colors, icons and so on)."

ENLARGE

This puzzle from 'Taking Sudoku Seriously' has as many empty rows, columns and 3-by-3 blocks as mathematically possible while still having only one solution.

Jason Rosenhouse and Laura Taalman quote that statement on page 5 of "Taking Sudoku Seriously" in order to promptly refute it. As well they should. For in its essence Sudoku is 100% mathematical. What Mr. Delahaye probably meant was that the puzzle is not arithmetical. But arithmetic—addition, subtraction, multiplication and division, the stuff familiar from most people's earliest, and often only, exposure to math—is just one tiny part of the discipline.

The misconception that mathematics and arithmetic are the same is a common one. By their own declaration, Mr. Rosenhouse and Ms. Taalman wrote "Taking Sudoku Seriously" in large part to overturn that false view, and to give readers a more accurate sense of what mathematics really is.

Taking Sudoku Seriously

By Jason Rosenhouse and Laura Taalman

Oxford, 214 pages, $21.95

In seeking to explain to the general public what mathematicians do, the authors (both math professors themselves) could have written the kind of book I have become known for, where I explain, in as understandable a manner as I can, the kinds of mathematics we professionals do. They chose instead to develop two parallel themes. In one, they use mathematics—algebraic-looking stuff with formulas, equations, and numbers that are used to count things—to analyze Sudoku. In the other, they use Sudoku to explain the true nature of mathematics (or perhaps I should say mathematical thinking, since they obviously do not cover much of the vast content of modern mathematics).

So rich is the Sudoku puzzle, that it is perfectly adequate to achieve that second goal. The authors show vividly that mathematics is really about the power of abstraction, the push to explain as much as possible in the most compact form possible. Numbers and arithmetic are a part of that enterprise, but there is a lot more besides. "Taking Sudoku Seriously" is an excellent vehicle whereby devotees of the puzzle can come to understand the nature of mathematics. Whether it will work in the other direction, by turning mathematicians into Sudoku players, is another matter.

If I am at all typical of mathematicians, the answer is no. The puzzle never really interested me. Not because I did not recognize at once its mathematical character. Rather, it didn't offer me anything I did not get from my day job. To me it was mathematics in significantly diminished form. It reminded me of the younger me who loved rock climbing but never enjoyed indoor climbing on artificial walls. Yes, the individual moves were the same, but the climbing wall lacked the grandeur and sense of reality—and truth be told the heightened senses aroused by the slight chance of death—of a genuine rock face.

In fact, I have never found puzzles satisfying; I always get a far greater thrill from mathematics, which has a rich and deep aesthetics, to say nothing of a huge importance to human life, that puzzles lack, though at a micro level the intellectual challenge is much the same. (Not all mathematicians share my indifference to puzzles, though many do.)

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Answer to top Sudoku.
Oxford University Press

Even when mathematicians pursue topics that seemingly bear no relationship to the messy world we all live in, their results frequently turn out to have important applications. For example, the ancient Greeks began a fascination with prime numbers that endures to this day, driven by nothing more than curiosity. Yet a theorem about prime numbers proved by the 17th-century Frenchman Pierre de Fermat (not his famous "Last Theorem", which he did not actually prove), seemingly having no connection to the everyday world, ultimately led to the encryption techniques that safeguard your password when you log on to your bank or make a purchase at an online bookstore.

Though Sudoku puzzles are a recent invention, they are a twist on a mathematical idea called a "Latin square" that goes back at least to Islamic thinkers around the 13th century. A Latin square of size n is an n-by-n square grid in which each row and each column contains every one of a set of numbers (1 to n, for example) or symbols. Often studied in a mystical or quasi-religious context, Latin squares were made an object of serious mathematical study by the great 18th-century Swiss mathematician Leonhard Euler, who posed a problem that wasn't solved until the beginning of the 20th century: Is it possible to arrange 36 officers, each having one of six different ranks and belonging to one of six different regiments, in a 6-by-6 square, so that each row and each file contains just one officer of each rank and just one from each regiment? (Euler, who suspected not, was eventually proved right.)

Latin squares were studied by Benjamin Franklin and other famous individuals for recreational purposes. In the 1930s, the influential British statistician R.A. Fisher used them to design an experiment to study crop yields. More recently, they have been used to design algorithms to encode digital files in a way that allows for the automatic detection and correction of errors that can arise during transmission over the Internet.

A Sudoku puzzle is just a 9-by-9 Latin square with an added restriction on the 3-by-3 blocks within it. There are 5,472,730,538 of them, it turns out—and many more if you count a puzzle that has been rotated 90 degrees or flipped left-to-right as distinct from the original. That finding draws on group theory and counting techniques, two of the many mathematical topics that Mr. Rosenhouse and Ms. Taalman cover in their analysis of Sudoku, along with searching techniques, graph theory and algebra (polynomials and all).

But will all this help puzzlers improve their Sudoku play? I doubt it, and the authors make no such claim. What enthusiasts may get is a deeper appreciation for the puzzle they love. It is just because Sudoku has mathematical depth that it can be so addictive and so satisfying. Though dull, uninspired teaching can reduce mathematics to tedium, the discipline was created over thousands of years by human minds seeking playful mental challenges that could be justified by their benefits (both direct and indirect) to society.

Sudoku players may also get something else: a realization that they are actually good mathematical thinkers—they really can do math. As for someone interested in neither Sudoku nor mathematics? Well, such a person will surely never acquire or open this book.

—Mr. Devlin is a mathematician at Stanford. His latest book is "The Man of Numbers: Fibonacci's Arithmetic Revolution."

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